Eleven chairs are numbered 1 through 11. Four girls and seven boys sit on these chairs at random. What is the probability that chair 5 is occupied by a boy? Solution The answer is just 7/11. At random allows us to just consider any seat (it doesn\'t matter if it is the 5th 6th or 11th seat)..

1. Eleven friends are about to seat themselves at a round table for a nice spaghetti dinner. The
following arrangements must be made: Charlie, Mac, and Dennis need to sit next to each other.
Liam, Jack and Ryan need to sit next to each other. Frank and Artemis need to sit next to each
other. None of the other guys have any seating preference. In how many ways can these eleven
friends arrange themselves around this table?
Solution
c,m,d must sit next to each other, 3! ways to arrange them, treat as one l,j,r " 3! " "
" f, a " 2! " " 3 other with no seating preference So the group acts like six groups. Since They
are seated in same order if the group is rotated, So pick 1 of 6, leaving 5, place them first, then
the others and their arrangements last ans=1*5*4*3*2*1*(3!)(3!)(2!) =5!*6*6*2=8640